We investigate the effect of an $\varepsilon$-room of perturbation tolerance on symmetric tensor decomposition. To be more precise, suppose a real symmetric $d$-tensor $f$, a norm $||.||$ on the space of symmetric $d$-tensors, and $\varepsilon >0$ are given. What is the smallest symmetric tensor rank in the $\varepsilon$-neighborhood of $f$? In other words, what is the symmetric tensor rank of $f$ after a clever $\varepsilon$-perturbation? We prove two theorems and develop three corresponding algorithms that give constructive upper bounds for this question. With expository goals in mind; we present probabilistic and convex geometric ideas behind our results, reproduce some known results, and point out open problems.

翻译：我们研究$\varepsilon$扰动容差对对称张量分解的影响。具体而言，给定实对称$d$阶张量$f$、对称$d$阶张量空间上的范数$||.||$以及$\varepsilon >0$，问：在$f$的$\varepsilon$邻域中，最小的对称张量秩是多少？换言之，经巧妙$\varepsilon$扰动后，$f$的对称张量秩是多少？我们证明两个定理，并开发三种相应算法，为该问题给出构造性上界。出于阐述目的，我们展示结果背后的概率与凸几何思想，复现若干已知结论，并指出未解决问题。